The normal state of an interacting Bose gas

This work analyses the thermodynamic properties of the normal (non-superfluid) state of a low density interacting Bose gas with a finite range interaction just above the superfluid transition temperature. The technique used to characterize this system is the cluster expansion developed by Lee and Yang, which was successfully applied to calculate the properties of a dilute normal Fermi gas near the ground state. The properties of a non-interacting Bose gas are reviewed in an unconventional manner by explicitly using the expansion developed by Robinson for the Bose integrals throughout. The sole half-integer power, branch point singularity in the Robinson expansion is seen to determine the properties of the gas at the Bose-Einstein condensation. The superfluid ground state of an interacting Bose gas is reviewed using the Bogoliubov transformation. The corrections from the cluster expansion to an interacting Bose gas are calculated up to second order in the s wave scattering length a.; The second order correction for the Bose gas consists of terms containing the Robinson expansions from the non-interacting Bose gas and a twelve-dimensional integral arising from the three-body interactions. This integral is reduced to three dimensions and calculated numerically, yielding a non-analytic term in an addition to those arising from the Robinson expansions. To within numerical precision this non-analytic term is equal to a power of one-half branch point singularity, similar to the Robinson expansion for the density. All these singularities make the cluster expansion a series of divergent terms at the transition temperature. All the singularities can be formally removed by performing a Legendre transformation to the Helmholtz free energy, resulting in an analytic series expansion. Consequently, finite corrections to second order in the interaction are obtained for the pressure, heat capacity, and velocity of sound at the transition temperature. The convergence property of the resulting series expansion for the Helmholtz free energy remains to be determined. Expansions for isobaric thermodynamic quantities consist of series with increasingly divergent terms at the transition temperature. Therefore, expansion of the interacting Bose system in terms of the non-interacting system gives divergent series and unphysical descriptions of an interacting Bose gas.